Hello,
(never mind what was there before)
Choose any . We have . Therefore, . Plug this result back into the second condition : . Now you might want to take a look at what the multiplicative inverses are in (it is special)

Sorry, I haven't had time to look at the other ones and it's bed time for me now
Quickly looking, maybe proving that there are no perfect squares of the form ? Maybe the prime factorization could help?